Find the Function Rule table[[x,y],[1,9],[2,18],[3,36]]

$\begin{array}{c} x & y \\ 1 & 9 \\ 2 & 18 \\ 3 & 36 \end{array}$

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $y = a x + b$ .

$9 = a (1) + b 18 = a (2) + b 36 = a (3) + b$

Calculate the values of $a$ and $b$ .

Solve for $a$ in $9 = a + b$ .

Rewrite the equation as $a + b = 9$ .

$a + b = 9$

$18 = a (2) + b$

$36 = a (3) + b$

Subtract $b$ from both sides of the equation.

$a = 9 - b$

$18 = a (2) + b$

$36 = a (3) + b$

$a = 9 - b$

$18 = a (2) + b$

$36 = a (3) + b$

Replace all occurrences of $a$ with $9 - b$ in each equation.

Replace all occurrences of $a$ in $18 = a (2) + b$ with $9 - b$ .

$18 = (9 - b) (2) + b$

$a = 9 - b$

$36 = a (3) + b$

Simplify the right side.

Simplify $(9 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$18 = 9 \cdot 2 - b \cdot 2 + b$

$a = 9 - b$

$36 = a (3) + b$

Multiply $9$ by $2$ .

$18 = 18 - b \cdot 2 + b$

$a = 9 - b$

$36 = a (3) + b$

Multiply $2$ by $- 1$ .

$18 = 18 - 2 b + b$

$a = 9 - b$

$36 = a (3) + b$

$18 = 18 - 2 b + b$

$a = 9 - b$

$36 = a (3) + b$

Add $- 2 b$ and $b$ .

$18 = 18 - b$

$a = 9 - b$

$36 = a (3) + b$

$18 = 18 - b$

$a = 9 - b$

$36 = a (3) + b$

$18 = 18 - b$

$a = 9 - b$

$36 = a (3) + b$

Replace all occurrences of $a$ in $36 = a (3) + b$ with $9 - b$ .

$36 = (9 - b) (3) + b$

$18 = 18 - b$

$a = 9 - b$

Simplify the right side.

Simplify $(9 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$36 = 9 \cdot 3 - b \cdot 3 + b$

$18 = 18 - b$

$a = 9 - b$

Multiply $9$ by $3$ .

$36 = 27 - b \cdot 3 + b$

$18 = 18 - b$

$a = 9 - b$

Multiply $3$ by $- 1$ .

$36 = 27 - 3 b + b$

$18 = 18 - b$

$a = 9 - b$

$36 = 27 - 3 b + b$

$18 = 18 - b$

$a = 9 - b$

Add $- 3 b$ and $b$ .

$36 = 27 - 2 b$

$18 = 18 - b$

$a = 9 - b$

$36 = 27 - 2 b$

$18 = 18 - b$

$a = 9 - b$

$36 = 27 - 2 b$

$18 = 18 - b$

$a = 9 - b$

$36 = 27 - 2 b$

$18 = 18 - b$

$a = 9 - b$

Solve for $b$ in $36 = 27 - 2 b$ .

Rewrite the equation as $27 - 2 b = 36$ .

$27 - 2 b = 36$

$18 = 18 - b$

$a = 9 - b$

Move all terms not containing $b$ to the right side of the equation.

Subtract $27$ from both sides of the equation.

$- 2 b = 36 - 27$

$18 = 18 - b$

$a = 9 - b$

Subtract $27$ from $36$ .

$- 2 b = 9$

$18 = 18 - b$

$a = 9 - b$

$- 2 b = 9$

$18 = 18 - b$

$a = 9 - b$

Divide each term in $- 2 b = 9$ by $- 2$ and simplify.

Divide each term in $- 2 b = 9$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{9}{- 2}$

$18 = 18 - b$

$a = 9 - b$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{9}{- 2}$

$18 = 18 - b$

$a = 9 - b$

Divide $b$ by $1$ .

$b = \frac{9}{- 2}$

$18 = 18 - b$

$a = 9 - b$

$b = \frac{9}{- 2}$

$18 = 18 - b$

$a = 9 - b$

$b = \frac{9}{- 2}$

$18 = 18 - b$

$a = 9 - b$

Simplify the right side.

Move the negative in front of the fraction.

$b = - \frac{9}{2}$

$18 = 18 - b$

$a = 9 - b$

$b = - \frac{9}{2}$

$18 = 18 - b$

$a = 9 - b$

$b = - \frac{9}{2}$

$18 = 18 - b$

$a = 9 - b$

$b = - \frac{9}{2}$

$18 = 18 - b$

$a = 9 - b$

Replace all occurrences of $b$ with $- \frac{9}{2}$ in each equation.

Replace all occurrences of $b$ in $18 = 18 - b$ with $- \frac{9}{2}$ .

$18 = 18 - (- \frac{9}{2})$

$b = - \frac{9}{2}$

$a = 9 - b$

Simplify the right side.

Simplify $18 - (- \frac{9}{2})$ .

Multiply $- (- \frac{9}{2})$ .

Multiply $- 1$ by $- 1$ .

$18 = 18 + 1 (\frac{9}{2})$

$b = - \frac{9}{2}$

$a = 9 - b$

Multiply $\frac{9}{2}$ by $1$ .

$18 = 18 + \frac{9}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

$18 = 18 + \frac{9}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$18 = 18 \cdot \frac{2}{2} + \frac{9}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

Combine $18$ and $\frac{2}{2}$ .

$18 = \frac{18 \cdot 2}{2} + \frac{9}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

Combine the numerators over the common denominator.

$18 = \frac{18 \cdot 2 + 9}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

Simplify the numerator.

Multiply $18$ by $2$ .

$18 = \frac{36 + 9}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

Add $36$ and $9$ .

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = 9 - b$

Replace all occurrences of $b$ in $a = 9 - b$ with $- \frac{9}{2}$ .

$a = 9 - (- \frac{9}{2})$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Simplify the right side.

Simplify $9 - (- \frac{9}{2})$ .

Multiply $- (- \frac{9}{2})$ .

Multiply $- 1$ by $- 1$ .

$a = 9 + 1 (\frac{9}{2})$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Multiply $\frac{9}{2}$ by $1$ .

$a = 9 + \frac{9}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = 9 + \frac{9}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 9 \cdot \frac{2}{2} + \frac{9}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Combine $9$ and $\frac{2}{2}$ .

$a = \frac{9 \cdot 2}{2} + \frac{9}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Combine the numerators over the common denominator.

$a = \frac{9 \cdot 2 + 9}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Simplify the numerator.

Multiply $9$ by $2$ .

$a = \frac{18 + 9}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Add $18$ and $9$ .

$a = \frac{27}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = \frac{27}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = \frac{27}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = \frac{27}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

$a = \frac{27}{2}$

$18 = \frac{45}{2}$

$b = - \frac{9}{2}$

Since $18 = \frac{45}{2}$ is not true, there is no solution.

No solution

Since $y \neq y$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Check if the function rule is quadratic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Build a set of $3$ equations from the table such that $y = a x^{2} + b x + c$ .

Calculate the values of $a$ , $b$ , and $c$ .

Solve for $a$ in $9 = a + b + c$ .

Rewrite the equation as $a + b + c = 9$ .

$a + b + c = 9$

$18 = a \cdot 2^{2} + b (2) + c$

$36 = a \cdot 3^{2} + b (3) + c$

Move all terms not containing $a$ to the right side of the equation.

Subtract $b$ from both sides of the equation.

$a + c = 9 - b$

$18 = a \cdot 2^{2} + b (2) + c$

$36 = a \cdot 3^{2} + b (3) + c$

Subtract $c$ from both sides of the equation.

$a = 9 - b - c$

$18 = a \cdot 2^{2} + b (2) + c$

$36 = a \cdot 3^{2} + b (3) + c$

$a = 9 - b - c$

$18 = a \cdot 2^{2} + b (2) + c$

$36 = a \cdot 3^{2} + b (3) + c$

$a = 9 - b - c$

$18 = a \cdot 2^{2} + b (2) + c$

$36 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ with $9 - b - c$ in each equation.

Replace all occurrences of $a$ in $18 = a \cdot 2^{2} + b (2) + c$ with $9 - b - c$ .

$18 = (9 - b - c) \cdot 2^{2} + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

Simplify $(9 - b - c) \cdot 2^{2} + b (2) + c$ .

Simplify each term.

Raise $2$ to the power of $2$ .

$18 = (9 - b - c) \cdot 4 + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Apply the distributive property.

$18 = 9 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Simplify.

Multiply $9$ by $4$ .

$18 = 36 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$18 = 36 - 4 b - c \cdot 4 + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$18 = 36 - 4 b - 4 c + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

$18 = 36 - 4 b - 4 c + b (2) + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Move $2$ to the left of $b$ .

$18 = 36 - 4 b - 4 c + 2 b + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

$18 = 36 - 4 b - 4 c + 2 b + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Simplify by adding terms.

Add $- 4 b$ and $2 b$ .

$18 = 36 - 2 b - 4 c + c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Add $- 4 c$ and $c$ .

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ in $36 = a \cdot 3^{2} + b (3) + c$ with $9 - b - c$ .

$36 = (9 - b - c) \cdot 3^{2} + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Simplify the right side.

Simplify $(9 - b - c) \cdot 3^{2} + b (3) + c$ .

Simplify each term.

Raise $3$ to the power of $2$ .

$36 = (9 - b - c) \cdot 9 + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Apply the distributive property.

$36 = 9 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Simplify.

Multiply $9$ by $9$ .

$36 = 81 - b \cdot 9 - c \cdot 9 + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Multiply $9$ by $- 1$ .

$36 = 81 - 9 b - c \cdot 9 + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Multiply $9$ by $- 1$ .

$36 = 81 - 9 b - 9 c + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = 81 - 9 b - 9 c + b (3) + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Move $3$ to the left of $b$ .

$36 = 81 - 9 b - 9 c + 3 b + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = 81 - 9 b - 9 c + 3 b + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Simplify by adding terms.

Add $- 9 b$ and $3 b$ .

$36 = 81 - 6 b - 9 c + c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Add $- 9 c$ and $c$ .

$36 = 81 - 6 b - 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = 81 - 6 b - 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = 81 - 6 b - 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = 81 - 6 b - 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$36 = 81 - 6 b - 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Solve for $b$ in $36 = 81 - 6 b - 8 c$ .

Rewrite the equation as $81 - 6 b - 8 c = 36$ .

$81 - 6 b - 8 c = 36$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Move all terms not containing $b$ to the right side of the equation.

Subtract $81$ from both sides of the equation.

$- 6 b - 8 c = 36 - 81$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Add $8 c$ to both sides of the equation.

$- 6 b = 36 - 81 + 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Subtract $81$ from $36$ .

$- 6 b = - 45 + 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 6 b = - 45 + 8 c$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Divide each term in $- 6 b = - 45 + 8 c$ by $- 6$ and simplify.

Divide each term in $- 6 b = - 45 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{- 45}{- 6} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{- 45}{- 6} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Divide $b$ by $1$ .

$b = \frac{- 45}{- 6} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{- 45}{- 6} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{- 45}{- 6} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 45$ and $- 6$ .

Factor $- 3$ out of $- 45$ .

$b = \frac{- 3 \cdot 15}{- 6} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Cancel the common factors.

Factor $- 3$ out of $- 6$ .

$b = \frac{- 3 \cdot 15}{- 3 \cdot 2} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Cancel the common factor.

$b = \frac{- 3 \cdot 15}{- 3 \cdot 2} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Rewrite the expression.

$b = \frac{15}{2} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} + \frac{8 c}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Cancel the common factor of $8$ and $- 6$ .

Factor $2$ out of $8 c$ .

$b = \frac{15}{2} + \frac{2 (4 c)}{- 6}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = \frac{15}{2} + \frac{2 (4 c)}{2 (- 3)}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Cancel the common factor.

$b = \frac{15}{2} + \frac{2 (4 c)}{2 \cdot - 3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Rewrite the expression.

$b = \frac{15}{2} + \frac{4 c}{- 3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} + \frac{4 c}{- 3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} + \frac{4 c}{- 3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Move the negative in front of the fraction.

$b = \frac{15}{2} - \frac{4 c}{3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$18 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Replace all occurrences of $b$ with $\frac{15}{2} - \frac{4 c}{3}$ in each equation.

Replace all occurrences of $b$ in $18 = 36 - 2 b - 3 c$ with $\frac{15}{2} - \frac{4 c}{3}$ .

$18 = 36 - 2 (\frac{15}{2} - \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Simplify the right side.

Simplify $36 - 2 (\frac{15}{2} - \frac{4 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$18 = 36 - 2 (\frac{15}{2}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$18 = 36 + 2 (- 1) (\frac{15}{2}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Cancel the common factor.

$18 = 36 + 2 \cdot (- 1 (\frac{15}{2})) - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Rewrite the expression.

$18 = 36 - 1 \cdot 15 - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 36 - 1 \cdot 15 - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Multiply $- 1$ by $15$ .

$18 = 36 - 15 - 2 (- \frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Multiply $- 2 (- \frac{4 c}{3})$ .

Multiply $- 1$ by $- 2$ .

$18 = 36 - 15 + 2 (\frac{4 c}{3}) - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Combine $2$ and $\frac{4 c}{3}$ .

$18 = 36 - 15 + \frac{2 (4 c)}{3} - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Multiply $4$ by $2$ .

$18 = 36 - 15 + \frac{8 c}{3} - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 36 - 15 + \frac{8 c}{3} - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 36 - 15 + \frac{8 c}{3} - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Subtract $15$ from $36$ .

$18 = 21 + \frac{8 c}{3} - 3 c$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

To write $- 3 c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$18 = 21 + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Simplify terms.

Combine $- 3 c$ and $\frac{3}{3}$ .

$18 = 21 + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Combine the numerators over the common denominator.

$18 = 21 + \frac{8 c - 3 c \cdot 3}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 21 + \frac{8 c - 3 c \cdot 3}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $8 c - 3 c \cdot 3$ .

Factor $c$ out of $8 c$ .

$18 = 21 + \frac{c \cdot 8 - 3 c \cdot 3}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Factor $c$ out of $- 3 c \cdot 3$ .

$18 = 21 + \frac{c \cdot 8 + c (- 3 \cdot 3)}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Factor $c$ out of $c \cdot 8 + c (- 3 \cdot 3)$ .

$18 = 21 + \frac{c (8 - 3 \cdot 3)}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 21 + \frac{c (8 - 3 \cdot 3)}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Multiply $- 3$ by $3$ .

$18 = 21 + \frac{c (8 - 9)}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Subtract $9$ from $8$ .

$18 = 21 + \frac{c \cdot - 1}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 21 + \frac{c \cdot - 1}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Move $- 1$ to the left of $c$ .

$18 = 21 + \frac{- 1 \cdot c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Move the negative in front of the fraction.

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - b - c$

Replace all occurrences of $b$ in $a = 9 - b - c$ with $\frac{15}{2} - \frac{4 c}{3}$ .

$a = 9 - (\frac{15}{2} - \frac{4 c}{3}) - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify the right side.

Simplify $9 - (\frac{15}{2} - \frac{4 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = 9 - \frac{15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply $- - \frac{4 c}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = 9 - \frac{15}{2} + 1 (\frac{4 c}{3}) - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply $\frac{4 c}{3}$ by $1$ .

$a = 9 - \frac{15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - \frac{15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = 9 - \frac{15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 9 \cdot \frac{2}{2} - \frac{15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Combine $9$ and $\frac{2}{2}$ .

$a = \frac{9 \cdot 2}{2} - \frac{15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{9 \cdot 2 - 15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify the numerator.

Multiply $9$ by $2$ .

$a = \frac{18 - 15}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Subtract $15$ from $18$ .

$a = \frac{3}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{4 c}{3} - c$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

To write $- c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = \frac{3}{2} + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify terms.

Combine $- c$ and $\frac{3}{3}$ .

$a = \frac{3}{2} + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{3}{2} + \frac{4 c - c \cdot 3}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{4 c - c \cdot 3}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $4 c - c \cdot 3$ .

Factor $c$ out of $4 c$ .

$a = \frac{3}{2} + \frac{c \cdot 4 - c \cdot 3}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Factor $c$ out of $- c \cdot 3$ .

$a = \frac{3}{2} + \frac{c \cdot 4 + c (- 1 \cdot 3)}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Factor $c$ out of $c \cdot 4 + c (- 1 \cdot 3)$ .

$a = \frac{3}{2} + \frac{c (4 - 1 \cdot 3)}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{c (4 - 1 \cdot 3)}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply $- 1$ by $3$ .

$a = \frac{3}{2} + \frac{c (4 - 3)}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Subtract $3$ from $4$ .

$a = \frac{3}{2} + \frac{c \cdot 1}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{c \cdot 1}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply $c$ by $1$ .

$a = \frac{3}{2} + \frac{c}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{c}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{c}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{c}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{3}{2} + \frac{c}{3}$

$18 = 21 - \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Solve for $c$ in $18 = 21 - \frac{c}{3}$ .

Rewrite the equation as $21 - \frac{c}{3} = 18$ .

$21 - \frac{c}{3} = 18$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Move all terms not containing $c$ to the right side of the equation.

Subtract $21$ from both sides of the equation.

$- \frac{c}{3} = 18 - 21$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Subtract $21$ from $18$ .

$- \frac{c}{3} = - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$- \frac{c}{3} = - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{c}{3}) = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 3 (- \frac{c}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{c}{3}$ into the numerator.

$- 3 \frac{- c}{3} = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- c}{3}) = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Cancel the common factor.

$3 \cdot (- 1 \frac{- c}{3}) = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Rewrite the expression.

$c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Multiply $c$ by $1$ .

$c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = - 3 \cdot - 3$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify the right side.

Multiply $- 3$ by $- 3$ .

$c = 9$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = 9$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = 9$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

$c = 9$

$a = \frac{3}{2} + \frac{c}{3}$

$b = \frac{15}{2} - \frac{4 c}{3}$

Replace all occurrences of $c$ with $9$ in each equation.

Replace all occurrences of $c$ in $a = \frac{3}{2} + \frac{c}{3}$ with $9$ .

$a = \frac{3}{2} + \frac{9}{3}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify the right side.

Simplify $\frac{3}{2} + \frac{9}{3}$ .

Divide $9$ by $3$ .

$a = \frac{3}{2} + 3$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \frac{3}{2} + 3 \cdot \frac{2}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

Combine $3$ and $\frac{2}{2}$ .

$a = \frac{3}{2} + \frac{3 \cdot 2}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{3 + 3 \cdot 2}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

Simplify the numerator.

Multiply $3$ by $2$ .

$a = \frac{3 + 6}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

Add $3$ and $6$ .

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - \frac{4 c}{3}$

Replace all occurrences of $c$ in $b = \frac{15}{2} - \frac{4 c}{3}$ with $9$ .

$b = \frac{15}{2} - \frac{4 (9)}{3}$

$a = \frac{9}{2}$

$c = 9$

Simplify the right side.

Simplify $\frac{15}{2} - \frac{4 (9)}{3}$ .

Simplify each term.

Cancel the common factor of $9$ and $3$ .

Factor $3$ out of $4 (9)$ .

$b = \frac{15}{2} - \frac{3 (4 \cdot (3))}{3}$

$a = \frac{9}{2}$

$c = 9$

Cancel the common factors.

Factor $3$ out of $3$ .

$b = \frac{15}{2} - \frac{3 (4 \cdot (3))}{3 (1)}$

$a = \frac{9}{2}$

$c = 9$

Cancel the common factor.

$b = \frac{15}{2} - \frac{3 (4 \cdot (3))}{3 \cdot 1}$

$a = \frac{9}{2}$

$c = 9$

Rewrite the expression.

$b = \frac{15}{2} - \frac{4 \cdot (3)}{1}$

$a = \frac{9}{2}$

$c = 9$

Divide $4 \cdot (3)$ by $1$ .

$b = \frac{15}{2} - (4 \cdot (3))$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - (4 \cdot (3))$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - (4 \cdot (3))$

$a = \frac{9}{2}$

$c = 9$

Multiply $4$ by $3$ .

$b = \frac{15}{2} - 1 \cdot 12$

$a = \frac{9}{2}$

$c = 9$

Multiply $- 1$ by $12$ .

$b = \frac{15}{2} - 12$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{15}{2} - 12$

$a = \frac{9}{2}$

$c = 9$

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = \frac{15}{2} - 12 \cdot \frac{2}{2}$

$a = \frac{9}{2}$

$c = 9$

Combine $- 12$ and $\frac{2}{2}$ .

$b = \frac{15}{2} + \frac{- 12 \cdot 2}{2}$

$a = \frac{9}{2}$

$c = 9$

Combine the numerators over the common denominator.

$b = \frac{15 - 12 \cdot 2}{2}$

$a = \frac{9}{2}$

$c = 9$

Simplify the numerator.

Multiply $- 12$ by $2$ .

$b = \frac{15 - 24}{2}$

$a = \frac{9}{2}$

$c = 9$

Subtract $24$ from $15$ .

$b = \frac{- 9}{2}$

$a = \frac{9}{2}$

$c = 9$

$b = \frac{- 9}{2}$

$a = \frac{9}{2}$

$c = 9$

Move the negative in front of the fraction.

$b = - \frac{9}{2}$

$a = \frac{9}{2}$

$c = 9$

$b = - \frac{9}{2}$

$a = \frac{9}{2}$

$c = 9$

$b = - \frac{9}{2}$

$a = \frac{9}{2}$

$c = 9$

$b = - \frac{9}{2}$

$a = \frac{9}{2}$

$c = 9$

List all of the solutions.

$b = - \frac{9}{2}, a = \frac{9}{2}, c = 9$

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{9}{2}$ , $b = - \frac{9}{2}$ , $c = 9$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = \frac{9}{2} \cdot 1 + (- \frac{9}{2}) \cdot (1) + 9$

Multiply $\frac{9}{2}$ by $1$ .

$y = \frac{9}{2} + (- \frac{9}{2}) \cdot (1) + 9$

Multiply $- 1$ by $1$ .

$y = \frac{9}{2} - \frac{9}{2} + 9$

Combine fractions.

Combine the numerators over the common denominator.

$y = 9 + \frac{9 - 9}{2}$

Simplify the expression.

Subtract $9$ from $9$ .

$y = 9 + \frac{0}{2}$

Divide $0$ by $2$ .

$y = 9 + 0$

Add $9$ and $0$ .

$y = 9$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 9$ and $y = 9$ .

$9 = 9$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{9}{2}$ , $b = - \frac{9}{2}$ , $c = 9$ , and $x = 2$ .

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of ${(2)}^{2}$ .

$y = \frac{9}{2} \cdot (2 \cdot 2) + (- \frac{9}{2}) \cdot (2) + 9$

Cancel the common factor.

$y = \frac{9}{2} \cdot (2 \cdot 2) + (- \frac{9}{2}) \cdot (2) + 9$

Rewrite the expression.

$y = 9 \cdot 2 + (- \frac{9}{2}) \cdot (2) + 9$

Multiply $9$ by $2$ .

$y = 18 + (- \frac{9}{2}) \cdot (2) + 9$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$y = 18 + \frac{- 9}{2} \cdot 2 + 9$

Cancel the common factor.

$y = 18 + \frac{- 9}{2} \cdot 2 + 9$

Rewrite the expression.

$y = 18 - 9 + 9$

Simplify by adding and subtracting.

Subtract $9$ from $18$ .

$y = 9 + 9$

Add $9$ and $9$ .

$y = 18$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 18$ and $y = 18$ .

$18 = 18$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{9}{2}$ , $b = - \frac{9}{2}$ , $c = 9$ , and $x = 3$ .

Simplify each term.

Raise $3$ to the power of $2$ .

$y = \frac{9}{2} \cdot 9 + (- \frac{9}{2}) \cdot (3) + 9$

Multiply $\frac{9}{2} \cdot 9$ .

Combine $\frac{9}{2}$ and $9$ .

$y = \frac{9 \cdot 9}{2} + (- \frac{9}{2}) \cdot (3) + 9$

Multiply $9$ by $9$ .

$y = \frac{81}{2} + (- \frac{9}{2}) \cdot (3) + 9$

Multiply $(- \frac{9}{2}) (3)$ .

Multiply $3$ by $- 1$ .

$y = \frac{81}{2} - 3 (\frac{9}{2}) + 9$

Combine $- 3$ and $\frac{9}{2}$ .

$y = \frac{81}{2} + \frac{- 3 \cdot 9}{2} + 9$

Multiply $- 3$ by $9$ .

$y = \frac{81}{2} + \frac{- 27}{2} + 9$

Move the negative in front of the fraction.

$y = \frac{81}{2} - \frac{27}{2} + 9$

Combine fractions.

Combine the numerators over the common denominator.

$y = 9 + \frac{81 - 27}{2}$

Simplify the expression.

Subtract $27$ from $81$ .

$y = 9 + \frac{54}{2}$

Divide $54$ by $2$ .

$y = 9 + 27$

Add $9$ and $27$ .

$y = 36$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 36$ and $y = 36$ .

$36 = 36$

Since $y = y$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Since all $y = y$ , the function is quadratic and follows the form $y = \frac{9 x^{2}}{2} - \frac{9 x}{2} + 9$ .

$y = \frac{9 x^{2}}{2} - \frac{9 x}{2} + 9$

Find the Function Rule table[[x,y],[1,9],[2,18],[3,36]]

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